FP2

Question 1

What is sin⍺cosβ equal to

Answer 1

0.5 (sin(⍺+β) + sin(⍺-β))

Question 2

What is sin(⍺)sin(β) equal to?

Answer 2

0.5(cos(⍺-β)-cos(⍺+β))

Question 3

What is cos(⍺)cos(β) equal to?

Answer 3

0.5(cos(⍺-β)+cos(⍺+β))

Question 4

in what form do you put cos²x for integration?

Answer 4

(cos 2x +1) /2

Question 5

in what form do you put sin²x for integration?

Answer 5

(1-cos2x)/2

Question 6

what is arccot(x) equal to?

Answer 6

arctan(1/x).

true for other trig functions.

Question 7

What is x equal to when converting to polar form?

Answer 7

r*cos theta

Question 8

what is sinθ equal to when converting to Cartesian form?

Answer 8

y/r

Question 9

what is x² +y² equal to when converting to polar form

Answer 9

r²

Question 10

what is tanθ equal to when converting to polar form

Answer 10

y/x

Question 11

how is the area of a polar graph found?

Answer 11

intergrate 0.5 r²

Question 12

What does r=a*secθ represent?

Answer 12

a straight vertical line

Question 13

what does r=a*cosecθ represent

Answer 13

a straight horizontal line

Question 14

what is the general solution to sinx=y

Answer 14

x=nπ+(-1)^n*arcsin(y)

Question 15

what is the general solution to cosx=y

Answer 15

x=2nπ±arcos(y)

Question 16

what is the general solution to tan(x)=y

Answer 16

x=nπ+arctan(y)

Question 17

what is the principle polar co-ordinate ?

Answer 17

r is greater than 0 and -π

Question 18

What is tanh(x) equal to?

Answer 18

Sinh(x)/cosh(x)

Question 19

What is coth x

Answer 19

1/ tanh x

Question 20

What is sech x

Answer 20

1/cosh x

Question 21

What is cosech x

Answer 21

1/sinh x

Question 22

If you have the matrices A and B and the in verses of each, what is (AB)⁻¹ equal to?

Answer 22

(AB)⁻¹ = B⁻¹ * A⁻¹

Question 23

What are the 2 identities used to lower the powers of hyperbolic functions?

Answer 23

2sinh²(x) = cosh(2x)-1
2cosh²(x)= cosh(2x)+1

Question 24

What is the trace of a matrix and how does this relate to eigenvectors?

Answer 24

The trace is the sum of the leading diagonal and this equals the sum of the eigenvectors.

Question 25

How do you express a+bi in the form re^i⊝?

Answer 25

r is |a+bi|

⊝= arg(a+bi)

Question 26

What is polar form?

Answer 26

Z=r(cos⊝+isin⊝)

Question 27

How do you multiply numbers in polar form?

Answer 27

r of the product is the r₁×r₂

⊝ of the product = ⊝₁+⊝₂

Question 28

What is de moivre’s theorem?

Answer 28

(Cos⊝+isin⊝)ª= cos a⊝ +isin a⊝

Question 29

What is arg(iz)?

Answer 29

Arg(z) +0.5π

Question 30

What happens to the determinant if you swap two columns?

Answer 30

The sign is reversed.

Question 31

How does a cyclic interchange affect the determinant?

Answer 31

It doesn’t

Question 32

If a matrix has two identical columns, what is the determinant?

Answer 32

0

Question 33

If you multiply one column of a matrix by K, what happens to the determinant?

Answer 33

Multiplied by K

If you multiply every column by a constant the determinant is multiplied by all the constants.

Question 34

What is adj(M)?

Answer 34

The inverse of the matrix but with out being divided by the determinant

Question 35

What is a cheaf ?

Answer 35

Line where three planes meet

Question 36

What is the relationship between eigenvectors and eigenvalues.

Answer 36

(M^n)* vector =(value^n)*vector

Question 37

How do you find eigen values?

Answer 37

Find the solutions to the characteristic equation.

Question 38

How do you find eigen vectors from a characteristic equation?

Answer 38

Subtract the eigen value from the leading diagonal
Multiply be x,y,z
Rearrange the equations.

Question 39

How do you find S and ¥ such that S¥S^-1 =M

Answer 39

Find eigen values for M
Find corresponding vectors
Put the vectors together for S
Put values in diagonal for ¥
Make sure vectors and values are in same order.

Question 40

If you have to find M to a power but know ¥ and S, what do you do?

Answer 40

Just put ¥ to that power and multiply by S and S^-1

Question 41

How is the determinant of a matrix found from the characteristic equation?

Answer 41

It is the negative of the constant at the end provided the x³ term is positive.

Question 42

What substitution would you use to integrate 1÷ (a+bx²)

Answer 42

X(b)^0.5 = tan(theta)

Question 43

What substition would you use to integrate (a² -x²)^0.5

Answer 43

X=asin(theta)

Question 44

What substition would you use to integrate (a²+x²)^0.5?

Answer 44

x =a sinh

Question 45

What substition would you use to integrate (x²-a²)^0.5

Answer 45

X= a*cosh

Question 46

What substitution would you use to intergrate (x²+1)^0.5

Answer 46

U= sinh(x)

Question 47

What would you substitute intergrate (a²-x²)^-1.5 ?

Answer 47

X=2sin theta

X=2cos theta